Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods

Courtiel, Julien; Fusy, Éric; Lepoutre, Mathias; Mishna, Marni

doi:10.1016/j.ejc.2017.10.003

Cited by 8 publications

(13 citation statements)

References 28 publications

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“…This is somehow reminiscent of [6,5]. We wonder whether there are some other examples of this phenomenon, or even a generic framework for such bijections.…”

Section: Resultsmentioning

confidence: 95%

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

Courtiel

Price

Marcovici

2021

Electron. J. Combin.

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This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections. Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

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“…This is somehow reminiscent of [6,5]. We wonder whether there are some other examples of this phenomenon, or even a generic framework for such bijections.…”

Section: Resultsmentioning

confidence: 95%

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

Courtiel

Price

Marcovici

2021

Electron. J. Combin.

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“…• The one between triangular paths and Motzkin paths transform two-dimensional walks with no constraint on the endpoint into one-dimensional walks which must finish at the origin; • the one between pyramid paths and waffle walks transform three-dimensional walks with no constraint on the endpoint into two-dimensional walks which must end on one of the axis. This is somehow reminiscent of [6,5]. We wonder whether there are some other examples of this phenomenon, or even a generic framework for such bijections.…”

Section: Discussionmentioning

confidence: 95%

Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude

Courtiel¹,

Price²,

Marcovici³

2020

Preprint

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This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0 • , 60 • , 120 • , 180 • , 240 • , 300 • ) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height.We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2 n -to-one function from generic triangular walks to triangular walks only using directions 0 • , 120 • , 240 • . The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections.Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

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“…This is immediately reminiscent of the generating function formulas in Equation ( 6). Several authors have made the connection, in particular, Gessel, Weinstein and Wilf [17], Zeilberger [43], Xin [42], Eu et al [14,15], Burrill et al [9] and Courtiel et al [12]. In almost every case there is a natural parameter which is equidistributed with number of odd columns in the tableaux.…”

Section: Lattice Walk Modelsmentioning

confidence: 99%

“…Theorem 4 (Courtiel, Fusy, Lepoutre and Mishna 2018 [12,Theorem 20]). For k ≥ 1 and n ≥ 0, there is an explicit bijection between simple axis-walks of length n staying in W C (k) and simple excursions of length n staying in W D (k), starting from ( 1 2 , .…”

Section: 4mentioning

confidence: 99%

See 1 more Smart Citation

On Standard Young Tableaux of Bounded Height

Mishna

2019

Association for Women in Mathematics Series

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Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods

Cited by 8 publications

References 28 publications

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude

On Standard Young Tableaux of Bounded Height

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